A336512
Total sum of the left-to-right minima in all compositions of n.
Original entry on oeis.org
0, 1, 3, 8, 17, 38, 78, 162, 330, 672, 1355, 2736, 5503, 11058, 22191, 44507, 89198, 178697, 357852, 716440, 1434041, 2869935, 5742801, 11490298, 22988084, 45988166, 91995547, 184021931, 368093352, 736266262, 1472660452, 2945526806, 5891385159, 11783304479
Offset: 0
a(4) = 1 + 1 + 1 + 2 + 1 + 2 + 1 + 3 + 1 + 4 = 17: (1)111, (1)12, (1)21, (2)(1)1, (2)2, (1)3, (3)(1), (4).
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b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> [0,
`if`(j b(n, n+1)[2]:
seq(a(n), n=0..50);
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b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {0,
If[j < m, j*p[[1]], 0]} + p][b[n - j, Min[m, j]]], {j, 1, n}]];
a[n_] := b[n, n + 1][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 07 2022, after Alois P. Heinz *)
A336718
Total number of left-to-right maxima in all compositions of n into distinct parts.
Original entry on oeis.org
0, 1, 1, 4, 4, 7, 18, 21, 32, 46, 107, 121, 193, 257, 379, 728, 900, 1299, 1806, 2529, 3360, 6182, 7387, 10807, 14385, 20217, 26207, 36450, 58194, 72887, 101130, 135379, 183178, 240796, 323307, 417625, 649959, 797623, 1096645, 1426108, 1931340, 2470541
Offset: 0
a(6) = 18 = 3+2+2+2+1+1+2+1+2+1+1: (1)(2)(3), (1)(3)2, (2)1(3), (2)(3)1, (3)12, (3)21, (2)(4), (4)2, (1)(5), (5)1, (6).
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g:= proc(n) option remember;
`if`(n<2, n, (2*n-1)*g(n-1)-(n-1)^2*g(n-2))
end:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..50);
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g[n_] := g[n] = If[n < 2, n, (2 n - 1) g[n - 1] - (n - 1)^2 g[n - 2]];
b[n_, i_, p_] := b[n, i, p] = If[i (i + 1)/2 < n, 0, If[n == 0, g[p], b[n, i - 1, p] + b[n - i, Min[n - i, i - 1], p + 1]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)
A336771
Total sum of the left-to-right maxima in all compositions of n into distinct parts.
Original entry on oeis.org
0, 1, 2, 8, 11, 22, 53, 75, 123, 193, 418, 538, 894, 1268, 1950, 3567, 4799, 7143, 10355, 14968, 20701, 36398, 46420, 69071, 94972, 136385, 182522, 259104, 402405, 527090, 741569, 1015491, 1397661, 1880541, 2567202, 3392612, 5153156, 6553844, 9088372, 12040797
Offset: 0
a(6) = 53 = 6+4+5+5+3+3+6+4+6+5+6: (1)(2)(3), (1)(3)2, (2)1(3), (2)(3)1, (3)12, (3)21, (2)(4), (4)2, (1)(5), (5)1, (6).
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b:= proc(n, i, k, m) option remember; `if`(i
(2*i-k+1)*k/2, 0, `if`(n=0, [1, 0], b(n, i-1, k, m)+
(p-> p+[0, p[1]*i/(m+1-k)])(b(n-i, min(n-i, i-1), k-1, m))))
end:
a:= n-> add(b(n$2, k$2)[2]*k!, k=1..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..40);
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b[n_, i_, k_, m_] := b[n, i, k, m] = If[i < k || n >
(2*i - k + 1)*k/2, {0, 0}, If[n == 0, {1, 0}, b[n, i - 1, k, m] +
Function[p, p+{0, p[[1]]*i/(m+1-k)}][b[n-i, Min[n-i, i-1], k-1, m]]]];
a[n_] := Sum[b[n, n, k, k][[2]]*k!, {k, 1, Floor[(Sqrt[8*n + 1] - 1)/2]}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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